Pertanyaan

Find the value of the following limit. b. x → 0 lim [ ( x − 2 1 ) 2 − ( 2 x + x 2 1 ) ]

Find the value of the following limit.

b. $x \to 0 lim [(x - \frac{1}{2})^{2} - (2 x + \frac{1}{x ^{2}})]$

D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Jawaban

nilai dari adalah

nilai dari begin mathsize 14px style limit as x rightwards arrow 0 of open square brackets open parentheses x minus 1 half close parentheses squared minus open parentheses 2 x plus 1 over x squared close parentheses close square brackets end style

begin mathsize 14px style limit as x rightwards arrow 0 of open square brackets open parentheses x minus 1 half close parentheses squared minus open parentheses 2 x plus 1 over x squared close parentheses close square brackets end style

adalah

Pembahasan

Selesaikan limit tersebut dengan metode substitusiterlebih dahulu. Karena jika disubtitusikan secara langsungmenghasilkan bentuk tak tentu maka gunakan metode lainuntuk menyelesaikan limit tersebut Ubah bentuk limittersebut menjadi bentuk Sehingga Ingat kembali bahwa Maka Jadi, nilai dari adalah

Selesaikan limit tersebut dengan metode substitusi terlebih dahulu.

Karena jika disubtitusikan secara langsung menghasilkan bentuk tak tentu maka gunakan metode lain untuk menyelesaikan limit tersebut

Ubah bentuk limit tersebut menjadi bentuk $limit as x rightwards arrow c of space fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction$

Sehingga

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of open square brackets open parentheses x minus 1 half close parentheses squared minus open parentheses 2 x plus 1 over x squared close parentheses close square brackets end cell equals cell limit as x rightwards arrow 0 of open square brackets open parentheses x squared minus x plus 1 fourth close parentheses minus open parentheses fraction numerator 2 x cubed plus 1 over denominator x squared end fraction close parentheses close square brackets end cell row blank equals cell limit as x rightwards arrow 0 of open square brackets fraction numerator x squared left parenthesis x squared minus x plus begin display style 1 fourth end style right parenthesis minus 2 x cubed plus 1 over denominator x squared end fraction close square brackets end cell row blank equals cell limit as x rightwards arrow 0 of space open square brackets fraction numerator x to the power of 4 minus x cubed plus begin display style 1 fourth x squared end style minus 2 x cubed plus 1 over denominator x squared end fraction close square brackets end cell row blank equals cell limit as x rightwards arrow 0 of space open square brackets fraction numerator x to the power of 4 minus 3 x cubed plus begin display style 1 fourth x squared end style plus 1 over denominator x squared end fraction close square brackets end cell end table end style$

Ingat kembali bahwa

$limit as x rightwards arrow c of space fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction equals limit as x rightwards arrow c of space fraction numerator f apostrophe apostrophe left parenthesis x right parenthesis over denominator g apostrophe apostrophe left parenthesis x right parenthesis end fraction$

Maka

$table attributes columnalign right center left columnspacing 0px end attributes row cell size 12px lim with size 12px x size 12px rightwards arrow size 12px 0 below size 12px space begin mathsize 12px style fraction numerator x to the power of 4 minus 3 x cubed plus begin display style 1 fourth x squared end style plus 1 over denominator x squared end fraction end style end cell size 12px equals cell size 12px lim with size 12px x size 12px rightwards arrow size 12px 0 below size 12px space begin mathsize 12px style fraction numerator 4 x cubed minus 9 x squared plus begin display style 1 half x end style over denominator 2 x end fraction end style end cell row blank size 12px equals cell size 12px lim with size 12px x size 12px rightwards arrow size 12px 0 below size 12px space begin mathsize 12px style fraction numerator 12 x squared minus 18 x plus begin display style 1 half end style over denominator 2 end fraction end style end cell row blank size 12px equals cell fraction numerator size 12px 12 size 12px left parenthesis size 12px 0 size 12px right parenthesis to the power of size 12px 2 size 12px minus size 12px 12 size 12px left parenthesis size 12px 0 size 12px right parenthesis size 12px plus begin mathsize 12px display style 1 half end style over denominator size 12px 2 end fraction end cell row blank size 12px equals cell size 12px 1 over size 12px 4 end cell end table$

Jadi, nilai dari begin mathsize 14px style limit as x rightwards arrow 0 of open square brackets open parentheses x minus 1 half close parentheses squared minus open parentheses 2 x plus 1 over x squared close parentheses close square brackets end style adalah 1 fourth